Systems of Linear Equations Word Problems Worksheet | PDF - Free Printable

Let's solve each problem step by step using systems of linear equations.

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Problem 1:

> A multiple choice test consists of 100 questions. A correct answer is worth 2 marks, while an incorrect answer is worth -1 mark. If a student receives a score of 80, how many questions did this student answer incorrectly?

#### Step 1: Define variables
- Let $ x $ = number of correct answers
- Let $ y $ = number of incorrect answers

We know:
- Total questions: $ x + y = 100 $
- Score: $ 2x - 1y = 80 $

#### Step 2: Write the system
$$
\begin{cases}
x + y = 100 \\
2x - y = 80
\end{cases}
$$

#### Step 3: Solve
Add the two equations:
$$
(x + y) + (2x - y) = 100 + 80 \\
3x = 180 \Rightarrow x = 60
$$

Substitute into first equation:
$$
60 + y = 100 \Rightarrow y = 40
$$

✔ Answer: The student answered 40 questions incorrectly.

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Problem 2:

> White chocolate costs $2.00 per bar, and dark chocolate costs $2.50 per bar. If you buy 15 bars of chocolate for $34 dollars, how many bars of dark chocolate did you buy?

#### Step 1: Define variables
- Let $ w $ = number of white chocolate bars
- Let $ d $ = number of dark chocolate bars

We know:
- Total bars: $ w + d = 15 $
- Total cost: $ 2w + 2.5d = 34 $

#### Step 2: Write the system
$$
\begin{cases}
w + d = 15 \\
2w + 2.5d = 34
\end{cases}
$$

#### Step 3: Solve
From first equation: $ w = 15 - d $

Substitute into second:
$$
2(15 - d) + 2.5d = 34 \\
30 - 2d + 2.5d = 34 \\
30 + 0.5d = 34 \\
0.5d = 4 \Rightarrow d = 8
$$

✔ Answer: You bought 8 bars of dark chocolate.

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Problem 3:

> A cellular company’s revenue and cost functions are modeled by:
> $ R = 50n $, $ C = 10n + 300 $
> At what point will this company have no profit? (Profit = Revenue − Cost)

#### Step 1: Set Profit = 0
$$
R - C = 0 \\
50n - (10n + 300) = 0 \\
50n - 10n - 300 = 0 \\
40n = 300 \Rightarrow n = 7.5
$$

Since $ n $ represents number of phones sold, and it must be a whole number, but mathematically, no profit occurs at $ n = 7.5 $.

But in real-world terms, the break-even point is when profit is zero, so:

✔ Answer: The company has no profit when 7.5 phones are sold. (This means they break even between selling 7 and 8 phones.)

Note: In practical terms, they need to sell 8 phones to start making a profit.

But since the question asks for the point, we give the exact value.

✔ Answer: $ \boxed{n = 7.5} $ phones

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Problem 4:

> The sum of two numbers is 17. The difference between the larger number and the smaller number is 7. What is the value of the smaller number?

#### Step 1: Define variables
- Let $ x $ = smaller number
- Let $ y $ = larger number

We know:
- $ x + y = 17 $
- $ y - x = 7 $

#### Step 2: Solve
Add the two equations:
$$
(x + y) + (y - x) = 17 + 7 \\
2y = 24 \Rightarrow y = 12
$$

Then:
$$
x = 17 - y = 17 - 12 = 5
$$

✔ Answer: The smaller number is $ \boxed{5} $

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Problem 5:

> John has $5.05 in quarters and nickels in his pocket. If John only has 25 coins in his pocket, how many of the coins are quarters?

#### Step 1: Define variables
- Let $ q $ = number of quarters ($0.25 each)
- Let $ n $ = number of nickels ($0.05 each)

We know:
- Total coins: $ q + n = 25 $
- Total value: $ 0.25q + 0.05n = 5.05 $

#### Step 2: Write the system
$$
\begin{cases}
q + n = 25 \\
0.25q + 0.05n = 5.05
\end{cases}
$$

Multiply second equation by 100 to eliminate decimals:
$$
25q + 5n = 505
$$

Divide entire equation by 5:
$$
5q + n = 101
$$

Now use:
$$
q + n = 25 \\
5q + n = 101
$$

Subtract first from second:
$$
(5q + n) - (q + n) = 101 - 25 \\
4q = 76 \Rightarrow q = 19
$$

Then:
$$
n = 25 - 19 = 6
$$

✔ Answer: John has 19 quarters

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Problem 6:

> A class of 32 consists of students who either have red or yellow shirts. If there are 12 more people with red shirts than there are people with yellow shirts, how many of the students have red shirts?

#### Step 1: Define variables
- Let $ r $ = number with red shirts
- Let $ y $ = number with yellow shirts

We know:
- $ r + y = 32 $
- $ r = y + 12 $

#### Step 2: Substitute
$$
(y + 12) + y = 32 \\
2y + 12 = 32 \\
2y = 20 \Rightarrow y = 10
$$

Then:
$$
r = 10 + 12 = 22
$$

✔ Answer: 22 students have red shirts.

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✔ Final Answers Summary:

1. 40 questions answered incorrectly
2. 8 bars of dark chocolate
3. 7.5 phones sold (break-even point)
4. 5 is the smaller number
5. 19 quarters
6. 22 students have red shirts

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